On the L$_\infty$ structure of Poisson gauge theory

TL;DR

This paper constructs an L$_ty^{full}$ algebra for Poisson gauge theory, providing explicit examples and insights into its structure, which can serve as a foundation for developing full non-commutative gauge theories.

Contribution

It introduces an explicit L$_ty^{full}$ algebra for Poisson gauge theory, satisfying homotopy relations and exploring its properties related to non-commutative gauge theories.

Findings

Constructed the minimal non-vanishing ll-brackets satisfying homotopy relations.

Provided explicit brackets for the first few levels of the full non-commutative gauge theory.

Showed that derivation properties hold only for canonical non-commutativity.

Abstract

The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an L$_\infty^{full}$ algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory. We derive the minimal set of non-vanishing $\ell$-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L$_\infty$ algebras. On the other hand, it can be used as a starting point for bootstrapping the full non-commutative gauge theory. The first few brackets of such a theory are constructed explicitly in the text. In addition we show that the derivation properties of $\ell$-brackets on L$_\infty^{full}$ with respect to the truncated product on the exterior algebra are satisfied only for the canonical non-commutativity. In general, L$_\infty^{full}$ does not have a structure of P$_\infty$ algebra.